Question

Indicate which of the following are exact numbers: (a) the mass of a 3 by 5 -inch index card, (b) the number of ounces in a pound, (c) the volume of a cup of Seattle's Best coffee, (d) the number of inches in a mile, (e) the number of microseconds in a week, (f) the number of pages in this book.

Short answer

The exact numbers among the given options are (b) the number of ounces in a pound (16 ounces), (d) the number of inches in a mile (63,360 inches), and (e) the number of microseconds in a week (604,800,000,000 microseconds). The other quantities (a), (c), and (f) are not exact numbers due to uncertainties or variations in their values.

Step-by-step solution

(a) Mass of a 3 by 5-inch index card

The mass of a 3 by 5-inch index card cannot be considered an exact number since it is a physical property of an object that could vary depending on factors like density, material, or manufacturing processes of the index card. Therefore, it is not an exact number.

(b) Number of ounces in a pound

The number of ounces in a pound is a well-defined value. There are 16 ounces in a pound, and this is an exact number as it is defined without any uncertainty involved.

(c) Volume of a cup of Seattle's Best coffee

The volume of a cup of Seattle's Best coffee can vary based on factors like how full the cup is or the size of cups used by the company. Therefore, this value has some uncertainty and is not an exact number.

(d) Number of inches in a mile

The number of inches in a mile is a defined value with no uncertainty. There are 63,360 inches in a mile. Therefore, this is an exact number.

(e) Number of microseconds in a week

The number of microseconds in a week is a defined value with no uncertainty. We can calculate this value as follows: There are 60 seconds in a minute, 60 minutes in an hour, 24 hours in a day, and 7 days in a week. Therefore, the total number of seconds in a week is: \[60 \times 60 \times 24 \times 7 = 604,800\space seconds\] Since there are \(1,000,000\) microseconds in one second, there are: \[604,800 \space seconds \times 1,000,000 \space \frac{microseconds}{second} = 604,800,000,000 \space microseconds\] As this value is obtained from defined quantities and values, the total number of microseconds in a week is an exact number.

(f) Number of pages in this book

The number of pages in a book is a directly counted quantity, which means that it is an exact number. There is no uncertainty involved in the counting of pages in a book, making it an exact value. In conclusion, (b), (d), and (e) are exact numbers, whereas (a), (c), and (f) are not.

Key concepts

Defined Values

Defined values are numbers that are established by definition, leaving no room for ambiguity or variation. These values form the foundation of precise calculations and are universally acknowledged. You can think of them as constants that do not change, regardless of context or condition.
For example, the number of ounces in a pound is a defined value. It stands reliably at 16 ounces per pound. This definition is set and recognized everywhere, making it an exact number. Similarly, the number of inches in a mile, which is 63,360, and the number of microseconds in a week, calculated by combining other constant-defined values, are exact numbers.
Defined values are crucial for consistency in mathematics and science, allowing for universally communicable and understandable data. They ensure that everyone who uses these values is on the same page, facilitating error-free computations.

Uncertainty in Measurements

In the world of science and everyday life, measurements often come with a certain degree of uncertainty. This uncertainty arises because measuring tools have limitations or because the physical state of what we measure isn't completely stable. When we talk about uncertain measurements, we're highlighting the potential small variations in values.
Consider the mass of a 3 by 5-inch index card. You might use scales to weigh it, but the outcome can depend on factors such as the scale's precision or slight inconsistencies in the paper's production. Similarly, the volume of a cup of coffee can vary based on how full the cup is filled, leading to slight differences each time you make the measurement.
Understanding uncertainty is important because it teaches us to be cautious and precise. It's about knowing that certain numbers may not be fixed like defined values, and we should account for possible fluctuations when using them in calculations.

Counting Quantities

Counting quantities refer to values obtained by direct counting without estimation or measurement tools. These are typically whole numbers, free from any uncertainty, because they involve enumeration. Counting is one of the simplest forms of quantitative analysis.
For instance, the number of pages in a book is a counting quantity. Once counted, this number remains fixed unless the book itself changes. Such values are considered exact because you determine them by a straightforward procedure: one by one enumeration of items.
Counting ensures precision in numbers, benefiting situations where exactness is required, such as inventories, population sizes, or document pages. The clarity and accuracy of counting quantities make them invaluable for clear and direct communications.